I have a domain registration open with a different company, and I would like to transfer it to Dynu without experiencing a lapse in service. Can I do that without waiting for the domain registration to expire?

We noticed that your domain name is already using our name servers. So you can simply place a domain transfer order. You may refer to https://www.dynu.com/Resources/Tutorials/DomainRegistration/TransferInDomainName
for detailed instructions.